Entanglement entropy in de Sitter: no pure states for conformal matter

In this paper, we consider the entanglement entropy of conformal matter for finite and semi-infinite entangling regions, as well as the formation of entanglement islands in four-dimensional de Sitter spacetime partially reduced to two dimensions. We analyze complementarity and pure state condition of entanglement entropy of pure states as a consistency test of the CFT formulas in this geometrical setup, which has been previously used in the literature to study the information paradox in higher-dimensional de Sitter in the context of the island proposal. We consider two different types of Cauchy surfaces in the extended static patch and flat coordinates, correspondingly. For former, we found that entanglement entropy of a pure state is always bounded from below by a constant and never becomes zero, as required by quantum mechanics. In turn, the difference between the entropies for some region and its complement, which should be zero for a pure state, in direct calculations essentially depends on how the boundaries of these regions evolve with time. Regarding the flat coordinates, it is impossible to regularize spacelike infinity in a way that would be compatible with complementarity and pure state condition, as opposed, for instance, to two-sided Schwarzschild black hole. Finally, we discuss the information paradox in de Sitter and show that the island formula does not resolve it, at least in this setup. Namely, we give examples of a region with a time-limited growth of entanglement entropy, for which there is no island solution, and the region, for which entanglement entropy does not grow, but the island solution exists.